LMFDB - Embedded newform 4012.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4012,2,Mod(1,4012)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4012, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4012.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4012 = 2^{2} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4012.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$32.0359812909$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	4012.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23607 q^{3} -1.00000 q^{5} +2.23607 q^{7} +2.00000 q^{9} +O(q^{10})$	$q-2.23607 q^{3} -1.00000 q^{5} +2.23607 q^{7} +2.00000 q^{9} -4.47214 q^{11} +1.23607 q^{13} +2.23607 q^{15} -1.00000 q^{17} +6.70820 q^{19} -5.00000 q^{21} -4.47214 q^{23} -4.00000 q^{25} +2.23607 q^{27} +1.00000 q^{29} +6.00000 q^{31} +10.0000 q^{33} -2.23607 q^{35} +1.70820 q^{37} -2.76393 q^{39} +1.47214 q^{41} -8.47214 q^{43} -2.00000 q^{45} -2.00000 q^{47} -2.00000 q^{49} +2.23607 q^{51} +0.527864 q^{53} +4.47214 q^{55} -15.0000 q^{57} +1.00000 q^{59} +13.7082 q^{61} +4.47214 q^{63} -1.23607 q^{65} +2.94427 q^{67} +10.0000 q^{69} -6.47214 q^{71} +13.4164 q^{73} +8.94427 q^{75} -10.0000 q^{77} -8.70820 q^{79} -11.0000 q^{81} -3.23607 q^{83} +1.00000 q^{85} -2.23607 q^{87} -2.47214 q^{89} +2.76393 q^{91} -13.4164 q^{93} -6.70820 q^{95} +5.70820 q^{97} -8.94427 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} + 4 q^{9}+O(q^{10}) $ 2 * q - 2 * q^5 + 4 * q^9	$ 2 q - 2 q^{5} + 4 q^{9} - 2 q^{13} - 2 q^{17} - 10 q^{21} - 8 q^{25} + 2 q^{29} + 12 q^{31} + 20 q^{33} - 10 q^{37} - 10 q^{39} - 6 q^{41} - 8 q^{43} - 4 q^{45} - 4 q^{47} - 4 q^{49} + 10 q^{53} - 30 q^{57} + 2 q^{59} + 14 q^{61} + 2 q^{65} - 12 q^{67} + 20 q^{69} - 4 q^{71} - 20 q^{77} - 4 q^{79} - 22 q^{81} - 2 q^{83} + 2 q^{85} + 4 q^{89} + 10 q^{91} - 2 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 + 4 * q^9 - 2 * q^13 - 2 * q^17 - 10 * q^21 - 8 * q^25 + 2 * q^29 + 12 * q^31 + 20 * q^33 - 10 * q^37 - 10 * q^39 - 6 * q^41 - 8 * q^43 - 4 * q^45 - 4 * q^47 - 4 * q^49 + 10 * q^53 - 30 * q^57 + 2 * q^59 + 14 * q^61 + 2 * q^65 - 12 * q^67 + 20 * q^69 - 4 * q^71 - 20 * q^77 - 4 * q^79 - 22 * q^81 - 2 * q^83 + 2 * q^85 + 4 * q^89 + 10 * q^91 - 2 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−2.23607	−1.29099	−0.645497	−	0.763763i	$-0.723350\pi$
$3$	−2.23607	−1.29099	−0.645497	+	0.763763i	$0.723350\pi$
$4$	0	0
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	0	0
$7$	2.23607	0.845154	0.422577	−	0.906327i	$-0.361126\pi$
$7$	2.23607	0.845154	0.422577	+	0.906327i	$0.361126\pi$
$8$	0	0
$9$	2.00000	0.666667
$10$	0	0
$11$	−4.47214	−1.34840	−0.674200	−	0.738549i	$-0.735511\pi$
$11$	−4.47214	−1.34840	−0.674200	+	0.738549i	$0.735511\pi$
$12$	0	0
$13$	1.23607	0.342824	0.171412	−	0.985199i	$-0.445167\pi$
$13$	1.23607	0.342824	0.171412	+	0.985199i	$0.445167\pi$
$14$	0	0
$15$	2.23607	0.577350
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	6.70820	1.53897	0.769484	−	0.638666i	$-0.220514\pi$
$19$	6.70820	1.53897	0.769484	+	0.638666i	$0.220514\pi$
$20$	0	0
$21$	−5.00000	−1.09109
$22$	0	0
$23$	−4.47214	−0.932505	−0.466252	−	0.884652i	$-0.654396\pi$
$23$	−4.47214	−0.932505	−0.466252	+	0.884652i	$0.654396\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	2.23607	0.430331
$28$	0	0
$29$	1.00000	0.185695	0.0928477	−	0.995680i	$-0.470403\pi$
$29$	1.00000	0.185695	0.0928477	+	0.995680i	$0.470403\pi$
$30$	0	0
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	0	0
$33$	10.0000	1.74078
$34$	0	0
$35$	−2.23607	−0.377964
$36$	0	0
$37$	1.70820	0.280827	0.140413	−	0.990093i	$-0.455157\pi$
$37$	1.70820	0.280827	0.140413	+	0.990093i	$0.455157\pi$
$38$	0	0
$39$	−2.76393	−0.442583
$40$	0	0
$41$	1.47214	0.229909	0.114955	−	0.993371i	$-0.463328\pi$
$41$	1.47214	0.229909	0.114955	+	0.993371i	$0.463328\pi$
$42$	0	0
$43$	−8.47214	−1.29199	−0.645994	−	0.763342i	$-0.723557\pi$
$43$	−8.47214	−1.29199	−0.645994	+	0.763342i	$0.723557\pi$
$44$	0	0
$45$	−2.00000	−0.298142
$46$	0	0
$47$	−2.00000	−0.291730	−0.145865	−	0.989305i	$-0.546597\pi$
$47$	−2.00000	−0.291730	−0.145865	+	0.989305i	$0.546597\pi$
$48$	0	0
$49$	−2.00000	−0.285714
$50$	0	0
$51$	2.23607	0.313112
$52$	0	0
$53$	0.527864	0.0725077	0.0362539	−	0.999343i	$-0.488458\pi$
$53$	0.527864	0.0725077	0.0362539	+	0.999343i	$0.488458\pi$
$54$	0	0
$55$	4.47214	0.603023
$56$	0	0
$57$	−15.0000	−1.98680
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	13.7082	1.75516	0.877578	−	0.479434i	$-0.159158\pi$
$61$	13.7082	1.75516	0.877578	+	0.479434i	$0.159158\pi$
$62$	0	0
$63$	4.47214	0.563436
$64$	0	0
$65$	−1.23607	−0.153315
$66$	0	0
$67$	2.94427	0.359700	0.179850	−	0.983694i	$-0.442439\pi$
$67$	2.94427	0.359700	0.179850	+	0.983694i	$0.442439\pi$
$68$	0	0
$69$	10.0000	1.20386
$70$	0	0
$71$	−6.47214	−0.768101	−0.384051	−	0.923312i	$-0.625471\pi$
$71$	−6.47214	−0.768101	−0.384051	+	0.923312i	$0.625471\pi$
$72$	0	0
$73$	13.4164	1.57027	0.785136	−	0.619324i	$-0.212593\pi$
$73$	13.4164	1.57027	0.785136	+	0.619324i	$0.212593\pi$
$74$	0	0
$75$	8.94427	1.03280
$76$	0	0
$77$	−10.0000	−1.13961
$78$	0	0
$79$	−8.70820	−0.979749	−0.489875	−	0.871793i	$-0.662957\pi$
$79$	−8.70820	−0.979749	−0.489875	+	0.871793i	$0.662957\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	−3.23607	−0.355205	−0.177602	−	0.984102i	$-0.556834\pi$
$83$	−3.23607	−0.355205	−0.177602	+	0.984102i	$0.556834\pi$
$84$	0	0
$85$	1.00000	0.108465
$86$	0	0
$87$	−2.23607	−0.239732
$88$	0	0
$89$	−2.47214	−0.262046	−0.131023	−	0.991379i	$-0.541826\pi$
$89$	−2.47214	−0.262046	−0.131023	+	0.991379i	$0.541826\pi$
$90$	0	0
$91$	2.76393	0.289739
$92$	0	0
$93$	−13.4164	−1.39122
$94$	0	0
$95$	−6.70820	−0.688247
$96$	0	0
$97$	5.70820	0.579580	0.289790	−	0.957090i	$-0.406414\pi$
$97$	5.70820	0.579580	0.289790	+	0.957090i	$0.406414\pi$
$98$	0	0
$99$	−8.94427	−0.898933
$100$	0	0
$101$	11.7082	1.16501	0.582505	−	0.812827i	$-0.302073\pi$
$101$	11.7082	1.16501	0.582505	+	0.812827i	$0.302073\pi$
$102$	0	0
$103$	−11.7082	−1.15364	−0.576822	−	0.816870i	$-0.695707\pi$
$103$	−11.7082	−1.15364	−0.576822	+	0.816870i	$0.695707\pi$
$104$	0	0
$105$	5.00000	0.487950
$106$	0	0
$107$	5.76393	0.557220	0.278610	−	0.960404i	$-0.410126\pi$
$107$	5.76393	0.557220	0.278610	+	0.960404i	$0.410126\pi$
$108$	0	0
$109$	0.472136	0.0452224	0.0226112	−	0.999744i	$-0.492802\pi$
$109$	0.472136	0.0452224	0.0226112	+	0.999744i	$0.492802\pi$
$110$	0	0
$111$	−3.81966	−0.362546
$112$	0	0
$113$	−2.76393	−0.260009	−0.130004	−	0.991513i	$-0.541499\pi$
$113$	−2.76393	−0.260009	−0.130004	+	0.991513i	$0.541499\pi$
$114$	0	0
$115$	4.47214	0.417029
$116$	0	0
$117$	2.47214	0.228549
$118$	0	0
$119$	−2.23607	−0.204980
$120$	0	0
$121$	9.00000	0.818182
$122$	0	0
$123$	−3.29180	−0.296811
$124$	0	0
$125$	9.00000	0.804984
$126$	0	0
$127$	5.76393	0.511466	0.255733	−	0.966747i	$-0.417683\pi$
$127$	5.76393	0.511466	0.255733	+	0.966747i	$0.417683\pi$
$128$	0	0
$129$	18.9443	1.66795
$130$	0	0
$131$	8.94427	0.781465	0.390732	−	0.920504i	$-0.372222\pi$
$131$	8.94427	0.781465	0.390732	+	0.920504i	$0.372222\pi$
$132$	0	0
$133$	15.0000	1.30066
$134$	0	0
$135$	−2.23607	−0.192450
$136$	0	0
$137$	−17.0000	−1.45241	−0.726204	−	0.687479i	$-0.758717\pi$
$137$	−17.0000	−1.45241	−0.726204	+	0.687479i	$0.758717\pi$
$138$	0	0
$139$	−0.944272	−0.0800921	−0.0400460	−	0.999198i	$-0.512750\pi$
$139$	−0.944272	−0.0800921	−0.0400460	+	0.999198i	$0.512750\pi$
$140$	0	0
$141$	4.47214	0.376622
$142$	0	0
$143$	−5.52786	−0.462263
$144$	0	0
$145$	−1.00000	−0.0830455
$146$	0	0
$147$	4.47214	0.368856
$148$	0	0
$149$	−14.6525	−1.20038	−0.600189	−	0.799858i	$-0.704908\pi$
$149$	−14.6525	−1.20038	−0.600189	+	0.799858i	$0.704908\pi$
$150$	0	0
$151$	−9.23607	−0.751621	−0.375810	−	0.926697i	$-0.622636\pi$
$151$	−9.23607	−0.751621	−0.375810	+	0.926697i	$0.622636\pi$
$152$	0	0
$153$	−2.00000	−0.161690
$154$	0	0
$155$	−6.00000	−0.481932
$156$	0	0
$157$	−22.1803	−1.77018	−0.885092	−	0.465416i	$-0.845905\pi$
$157$	−22.1803	−1.77018	−0.885092	+	0.465416i	$0.845905\pi$
$158$	0	0
$159$	−1.18034	−0.0936070
$160$	0	0
$161$	−10.0000	−0.788110
$162$	0	0
$163$	−3.05573	−0.239343	−0.119672	−	0.992814i	$-0.538184\pi$
$163$	−3.05573	−0.239343	−0.119672	+	0.992814i	$0.538184\pi$
$164$	0	0
$165$	−10.0000	−0.778499
$166$	0	0
$167$	−22.7082	−1.75721	−0.878607	−	0.477546i	$-0.841526\pi$
$167$	−22.7082	−1.75721	−0.878607	+	0.477546i	$0.841526\pi$
$168$	0	0
$169$	−11.4721	−0.882472
$170$	0	0
$171$	13.4164	1.02598
$172$	0	0
$173$	−4.18034	−0.317825	−0.158913	−	0.987293i	$-0.550799\pi$
$173$	−4.18034	−0.317825	−0.158913	+	0.987293i	$0.550799\pi$
$174$	0	0
$175$	−8.94427	−0.676123
$176$	0	0
$177$	−2.23607	−0.168073
$178$	0	0
$179$	−18.1803	−1.35886	−0.679431	−	0.733739i	$-0.737773\pi$
$179$	−18.1803	−1.35886	−0.679431	+	0.733739i	$0.737773\pi$
$180$	0	0
$181$	14.8885	1.10666	0.553328	−	0.832963i	$-0.313357\pi$
$181$	14.8885	1.10666	0.553328	+	0.832963i	$0.313357\pi$
$182$	0	0
$183$	−30.6525	−2.26590
$184$	0	0
$185$	−1.70820	−0.125590
$186$	0	0
$187$	4.47214	0.327035
$188$	0	0
$189$	5.00000	0.363696
$190$	0	0
$191$	4.94427	0.357755	0.178877	−	0.983871i	$-0.442753\pi$
$191$	4.94427	0.357755	0.178877	+	0.983871i	$0.442753\pi$
$192$	0	0
$193$	19.9443	1.43562	0.717810	−	0.696239i	$-0.245145\pi$
$193$	19.9443	1.43562	0.717810	+	0.696239i	$0.245145\pi$
$194$	0	0
$195$	2.76393	0.197929
$196$	0	0
$197$	19.8885	1.41700	0.708500	−	0.705711i	$-0.249372\pi$
$197$	19.8885	1.41700	0.708500	+	0.705711i	$0.249372\pi$
$198$	0	0
$199$	−17.7639	−1.25925	−0.629626	−	0.776898i	$-0.716792\pi$
$199$	−17.7639	−1.25925	−0.629626	+	0.776898i	$0.716792\pi$
$200$	0	0
$201$	−6.58359	−0.464371
$202$	0	0
$203$	2.23607	0.156941
$204$	0	0
$205$	−1.47214	−0.102818
$206$	0	0
$207$	−8.94427	−0.621670
$208$	0	0
$209$	−30.0000	−2.07514
$210$	0	0
$211$	−26.4721	−1.82242	−0.911208	−	0.411945i	$-0.864849\pi$
$211$	−26.4721	−1.82242	−0.911208	+	0.411945i	$0.864849\pi$
$212$	0	0
$213$	14.4721	0.991614
$214$	0	0
$215$	8.47214	0.577795
$216$	0	0
$217$	13.4164	0.910765
$218$	0	0
$219$	−30.0000	−2.02721
$220$	0	0
$221$	−1.23607	−0.0831469
$222$	0	0
$223$	−2.47214	−0.165546	−0.0827732	−	0.996568i	$-0.526378\pi$
$223$	−2.47214	−0.165546	−0.0827732	+	0.996568i	$0.526378\pi$
$224$	0	0
$225$	−8.00000	−0.533333
$226$	0	0
$227$	−20.6525	−1.37075	−0.685376	−	0.728189i	$-0.740362\pi$
$227$	−20.6525	−1.37075	−0.685376	+	0.728189i	$0.740362\pi$
$228$	0	0
$229$	3.70820	0.245045	0.122523	−	0.992466i	$-0.460902\pi$
$229$	3.70820	0.245045	0.122523	+	0.992466i	$0.460902\pi$
$230$	0	0
$231$	22.3607	1.47122
$232$	0	0
$233$	−3.23607	−0.212002	−0.106001	−	0.994366i	$-0.533805\pi$
$233$	−3.23607	−0.212002	−0.106001	+	0.994366i	$0.533805\pi$
$234$	0	0
$235$	2.00000	0.130466
$236$	0	0
$237$	19.4721	1.26485
$238$	0	0
$239$	−22.1246	−1.43112	−0.715561	−	0.698550i	$-0.753829\pi$
$239$	−22.1246	−1.43112	−0.715561	+	0.698550i	$0.753829\pi$
$240$	0	0
$241$	5.94427	0.382904	0.191452	−	0.981502i	$-0.438680\pi$
$241$	5.94427	0.382904	0.191452	+	0.981502i	$0.438680\pi$
$242$	0	0
$243$	17.8885	1.14755
$244$	0	0
$245$	2.00000	0.127775
$246$	0	0
$247$	8.29180	0.527594
$248$	0	0
$249$	7.23607	0.458567
$250$	0	0
$251$	−7.65248	−0.483020	−0.241510	−	0.970398i	$-0.577643\pi$
$251$	−7.65248	−0.483020	−0.241510	+	0.970398i	$0.577643\pi$
$252$	0	0
$253$	20.0000	1.25739
$254$	0	0
$255$	−2.23607	−0.140028
$256$	0	0
$257$	1.00000	0.0623783	0.0311891	−	0.999514i	$-0.490071\pi$
$257$	1.00000	0.0623783	0.0311891	+	0.999514i	$0.490071\pi$
$258$	0	0
$259$	3.81966	0.237342
$260$	0	0
$261$	2.00000	0.123797
$262$	0	0
$263$	−6.70820	−0.413646	−0.206823	−	0.978378i	$-0.566312\pi$
$263$	−6.70820	−0.413646	−0.206823	+	0.978378i	$0.566312\pi$
$264$	0	0
$265$	−0.527864	−0.0324264
$266$	0	0
$267$	5.52786	0.338300
$268$	0	0
$269$	−16.1803	−0.986533	−0.493266	−	0.869878i	$-0.664197\pi$
$269$	−16.1803	−0.986533	−0.493266	+	0.869878i	$0.664197\pi$
$270$	0	0
$271$	−7.18034	−0.436175	−0.218087	−	0.975929i	$-0.569982\pi$
$271$	−7.18034	−0.436175	−0.218087	+	0.975929i	$0.569982\pi$
$272$	0	0
$273$	−6.18034	−0.374051
$274$	0	0
$275$	17.8885	1.07872
$276$	0	0
$277$	−1.00000	−0.0600842	−0.0300421	−	0.999549i	$-0.509564\pi$
$277$	−1.00000	−0.0600842	−0.0300421	+	0.999549i	$0.509564\pi$
$278$	0	0
$279$	12.0000	0.718421
$280$	0	0
$281$	−25.3607	−1.51289	−0.756446	−	0.654057i	$-0.773066\pi$
$281$	−25.3607	−1.51289	−0.756446	+	0.654057i	$0.773066\pi$
$282$	0	0
$283$	−16.6525	−0.989887	−0.494943	−	0.868925i	$-0.664811\pi$
$283$	−16.6525	−0.989887	−0.494943	+	0.868925i	$0.664811\pi$
$284$	0	0
$285$	15.0000	0.888523
$286$	0	0
$287$	3.29180	0.194309
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−12.7639	−0.748235
$292$	0	0
$293$	−14.4164	−0.842216	−0.421108	−	0.907011i	$-0.638359\pi$
$293$	−14.4164	−0.842216	−0.421108	+	0.907011i	$0.638359\pi$
$294$	0	0
$295$	−1.00000	−0.0582223
$296$	0	0
$297$	−10.0000	−0.580259
$298$	0	0
$299$	−5.52786	−0.319685
$300$	0	0
$301$	−18.9443	−1.09193
$302$	0	0
$303$	−26.1803	−1.50402
$304$	0	0
$305$	−13.7082	−0.784929
$306$	0	0
$307$	−19.6525	−1.12163	−0.560813	−	0.827942i	$-0.689511\pi$
$307$	−19.6525	−1.12163	−0.560813	+	0.827942i	$0.689511\pi$
$308$	0	0
$309$	26.1803	1.48935
$310$	0	0
$311$	5.29180	0.300070	0.150035	−	0.988681i	$-0.452061\pi$
$311$	5.29180	0.300070	0.150035	+	0.988681i	$0.452061\pi$
$312$	0	0
$313$	−3.81966	−0.215900	−0.107950	−	0.994156i	$-0.534429\pi$
$313$	−3.81966	−0.215900	−0.107950	+	0.994156i	$0.534429\pi$
$314$	0	0
$315$	−4.47214	−0.251976
$316$	0	0
$317$	−3.52786	−0.198145	−0.0990723	−	0.995080i	$-0.531588\pi$
$317$	−3.52786	−0.198145	−0.0990723	+	0.995080i	$0.531588\pi$
$318$	0	0
$319$	−4.47214	−0.250392
$320$	0	0
$321$	−12.8885	−0.719368
$322$	0	0
$323$	−6.70820	−0.373254
$324$	0	0
$325$	−4.94427	−0.274259
$326$	0	0
$327$	−1.05573	−0.0583819
$328$	0	0
$329$	−4.47214	−0.246557
$330$	0	0
$331$	27.6525	1.51992	0.759959	−	0.649971i	$-0.225219\pi$
$331$	27.6525	1.51992	0.759959	+	0.649971i	$0.225219\pi$
$332$	0	0
$333$	3.41641	0.187218
$334$	0	0
$335$	−2.94427	−0.160863
$336$	0	0
$337$	−33.3050	−1.81424	−0.907118	−	0.420876i	$-0.861723\pi$
$337$	−33.3050	−1.81424	−0.907118	+	0.420876i	$0.861723\pi$
$338$	0	0
$339$	6.18034	0.335670
$340$	0	0
$341$	−26.8328	−1.45308
$342$	0	0
$343$	−20.1246	−1.08663
$344$	0	0
$345$	−10.0000	−0.538382
$346$	0	0
$347$	31.5967	1.69620	0.848101	−	0.529834i	$-0.177746\pi$
$347$	31.5967	1.69620	0.848101	+	0.529834i	$0.177746\pi$
$348$	0	0
$349$	−9.70820	−0.519668	−0.259834	−	0.965653i	$-0.583668\pi$
$349$	−9.70820	−0.519668	−0.259834	+	0.965653i	$0.583668\pi$
$350$	0	0
$351$	2.76393	0.147528
$352$	0	0
$353$	−0.763932	−0.0406600	−0.0203300	−	0.999793i	$-0.506472\pi$
$353$	−0.763932	−0.0406600	−0.0203300	+	0.999793i	$0.506472\pi$
$354$	0	0
$355$	6.47214	0.343505
$356$	0	0
$357$	5.00000	0.264628
$358$	0	0
$359$	−5.29180	−0.279290	−0.139645	−	0.990202i	$-0.544596\pi$
$359$	−5.29180	−0.279290	−0.139645	+	0.990202i	$0.544596\pi$
$360$	0	0
$361$	26.0000	1.36842
$362$	0	0
$363$	−20.1246	−1.05627
$364$	0	0
$365$	−13.4164	−0.702247
$366$	0	0
$367$	−6.65248	−0.347256	−0.173628	−	0.984811i	$-0.555549\pi$
$367$	−6.65248	−0.347256	−0.173628	+	0.984811i	$0.555549\pi$
$368$	0	0
$369$	2.94427	0.153273
$370$	0	0
$371$	1.18034	0.0612802
$372$	0	0
$373$	−8.47214	−0.438671	−0.219335	−	0.975650i	$-0.570389\pi$
$373$	−8.47214	−0.438671	−0.219335	+	0.975650i	$0.570389\pi$
$374$	0	0
$375$	−20.1246	−1.03923
$376$	0	0
$377$	1.23607	0.0636607
$378$	0	0
$379$	−6.70820	−0.344577	−0.172289	−	0.985047i	$-0.555116\pi$
$379$	−6.70820	−0.344577	−0.172289	+	0.985047i	$0.555116\pi$
$380$	0	0
$381$	−12.8885	−0.660300
$382$	0	0
$383$	−16.0000	−0.817562	−0.408781	−	0.912633i	$-0.634046\pi$
$383$	−16.0000	−0.817562	−0.408781	+	0.912633i	$0.634046\pi$
$384$	0	0
$385$	10.0000	0.509647
$386$	0	0
$387$	−16.9443	−0.861326
$388$	0	0
$389$	0.472136	0.0239382	0.0119691	−	0.999928i	$-0.496190\pi$
$389$	0.472136	0.0239382	0.0119691	+	0.999928i	$0.496190\pi$
$390$	0	0
$391$	4.47214	0.226166
$392$	0	0
$393$	−20.0000	−1.00887
$394$	0	0
$395$	8.70820	0.438157
$396$	0	0
$397$	−11.2361	−0.563922	−0.281961	−	0.959426i	$-0.590985\pi$
$397$	−11.2361	−0.563922	−0.281961	+	0.959426i	$0.590985\pi$
$398$	0	0
$399$	−33.5410	−1.67915
$400$	0	0
$401$	39.1246	1.95379	0.976895	−	0.213720i	$-0.0685580\pi$
$401$	39.1246	1.95379	0.976895	+	0.213720i	$0.0685580\pi$
$402$	0	0
$403$	7.41641	0.369438
$404$	0	0
$405$	11.0000	0.546594
$406$	0	0
$407$	−7.63932	−0.378667
$408$	0	0
$409$	−0.652476	−0.0322629	−0.0161314	−	0.999870i	$-0.505135\pi$
$409$	−0.652476	−0.0322629	−0.0161314	+	0.999870i	$0.505135\pi$
$410$	0	0
$411$	38.0132	1.87505
$412$	0	0
$413$	2.23607	0.110030
$414$	0	0
$415$	3.23607	0.158852
$416$	0	0
$417$	2.11146	0.103398
$418$	0	0
$419$	−10.1803	−0.497342	−0.248671	−	0.968588i	$-0.579994\pi$
$419$	−10.1803	−0.497342	−0.248671	+	0.968588i	$0.579994\pi$
$420$	0	0
$421$	6.00000	0.292422	0.146211	−	0.989253i	$-0.453292\pi$
$421$	6.00000	0.292422	0.146211	+	0.989253i	$0.453292\pi$
$422$	0	0
$423$	−4.00000	−0.194487
$424$	0	0
$425$	4.00000	0.194029
$426$	0	0
$427$	30.6525	1.48338
$428$	0	0
$429$	12.3607	0.596779
$430$	0	0
$431$	−8.94427	−0.430830	−0.215415	−	0.976523i	$-0.569110\pi$
$431$	−8.94427	−0.430830	−0.215415	+	0.976523i	$0.569110\pi$
$432$	0	0
$433$	−14.5279	−0.698165	−0.349082	−	0.937092i	$-0.613507\pi$
$433$	−14.5279	−0.698165	−0.349082	+	0.937092i	$0.613507\pi$
$434$	0	0
$435$	2.23607	0.107211
$436$	0	0
$437$	−30.0000	−1.43509
$438$	0	0
$439$	−26.8328	−1.28066	−0.640330	−	0.768100i	$-0.721202\pi$
$439$	−26.8328	−1.28066	−0.640330	+	0.768100i	$0.721202\pi$
$440$	0	0
$441$	−4.00000	−0.190476
$442$	0	0
$443$	12.4721	0.592569	0.296285	−	0.955100i	$-0.404252\pi$
$443$	12.4721	0.592569	0.296285	+	0.955100i	$0.404252\pi$
$444$	0	0
$445$	2.47214	0.117190
$446$	0	0
$447$	32.7639	1.54968
$448$	0	0
$449$	−31.3607	−1.48000	−0.740001	−	0.672606i	$-0.765175\pi$
$449$	−31.3607	−1.48000	−0.740001	+	0.672606i	$0.765175\pi$
$450$	0	0
$451$	−6.58359	−0.310009
$452$	0	0
$453$	20.6525	0.970338
$454$	0	0
$455$	−2.76393	−0.129575
$456$	0	0
$457$	25.7082	1.20258	0.601290	−	0.799031i	$-0.294654\pi$
$457$	25.7082	1.20258	0.601290	+	0.799031i	$0.294654\pi$
$458$	0	0
$459$	−2.23607	−0.104371
$460$	0	0
$461$	26.9443	1.25492	0.627460	−	0.778649i	$-0.284095\pi$
$461$	26.9443	1.25492	0.627460	+	0.778649i	$0.284095\pi$
$462$	0	0
$463$	−26.9443	−1.25221	−0.626103	−	0.779740i	$-0.715351\pi$
$463$	−26.9443	−1.25221	−0.626103	+	0.779740i	$0.715351\pi$
$464$	0	0
$465$	13.4164	0.622171
$466$	0	0
$467$	10.3607	0.479435	0.239718	−	0.970843i	$-0.422945\pi$
$467$	10.3607	0.479435	0.239718	+	0.970843i	$0.422945\pi$
$468$	0	0
$469$	6.58359	0.304002
$470$	0	0
$471$	49.5967	2.28530
$472$	0	0
$473$	37.8885	1.74212
$474$	0	0
$475$	−26.8328	−1.23117
$476$	0	0
$477$	1.05573	0.0483385
$478$	0	0
$479$	38.8328	1.77432	0.887158	−	0.461465i	$-0.152676\pi$
$479$	38.8328	1.77432	0.887158	+	0.461465i	$0.152676\pi$
$480$	0	0
$481$	2.11146	0.0962741
$482$	0	0
$483$	22.3607	1.01745
$484$	0	0
$485$	−5.70820	−0.259196
$486$	0	0
$487$	18.2361	0.826355	0.413178	−	0.910650i	$-0.364419\pi$
$487$	18.2361	0.826355	0.413178	+	0.910650i	$0.364419\pi$
$488$	0	0
$489$	6.83282	0.308991
$490$	0	0
$491$	38.2361	1.72557	0.862785	−	0.505571i	$-0.168718\pi$
$491$	38.2361	1.72557	0.862785	+	0.505571i	$0.168718\pi$
$492$	0	0
$493$	−1.00000	−0.0450377
$494$	0	0
$495$	8.94427	0.402015
$496$	0	0
$497$	−14.4721	−0.649164
$498$	0	0
$499$	−27.5410	−1.23291	−0.616453	−	0.787392i	$-0.711431\pi$
$499$	−27.5410	−1.23291	−0.616453	+	0.787392i	$0.711431\pi$
$500$	0	0
$501$	50.7771	2.26855
$502$	0	0
$503$	7.70820	0.343692	0.171846	−	0.985124i	$-0.445027\pi$
$503$	7.70820	0.343692	0.171846	+	0.985124i	$0.445027\pi$
$504$	0	0
$505$	−11.7082	−0.521008
$506$	0	0
$507$	25.6525	1.13927
$508$	0	0
$509$	−33.2361	−1.47316	−0.736581	−	0.676349i	$-0.763561\pi$
$509$	−33.2361	−1.47316	−0.736581	+	0.676349i	$0.763561\pi$
$510$	0	0
$511$	30.0000	1.32712
$512$	0	0
$513$	15.0000	0.662266
$514$	0	0
$515$	11.7082	0.515925
$516$	0	0
$517$	8.94427	0.393369
$518$	0	0
$519$	9.34752	0.410311
$520$	0	0
$521$	30.9443	1.35569	0.677847	−	0.735203i	$-0.262913\pi$
$521$	30.9443	1.35569	0.677847	+	0.735203i	$0.262913\pi$
$522$	0	0
$523$	22.2361	0.972315	0.486158	−	0.873871i	$-0.338398\pi$
$523$	22.2361	0.972315	0.486158	+	0.873871i	$0.338398\pi$
$524$	0	0
$525$	20.0000	0.872872
$526$	0	0
$527$	−6.00000	−0.261364
$528$	0	0
$529$	−3.00000	−0.130435
$530$	0	0
$531$	2.00000	0.0867926
$532$	0	0
$533$	1.81966	0.0788182
$534$	0	0
$535$	−5.76393	−0.249197
$536$	0	0
$537$	40.6525	1.75428
$538$	0	0
$539$	8.94427	0.385257
$540$	0	0
$541$	26.3607	1.13333	0.566667	−	0.823947i	$-0.308233\pi$
$541$	26.3607	1.13333	0.566667	+	0.823947i	$0.308233\pi$
$542$	0	0
$543$	−33.2918	−1.42869
$544$	0	0
$545$	−0.472136	−0.0202241
$546$	0	0
$547$	−15.0557	−0.643736	−0.321868	−	0.946784i	$-0.604311\pi$
$547$	−15.0557	−0.643736	−0.321868	+	0.946784i	$0.604311\pi$
$548$	0	0
$549$	27.4164	1.17010
$550$	0	0
$551$	6.70820	0.285779
$552$	0	0
$553$	−19.4721	−0.828039
$554$	0	0
$555$	3.81966	0.162136
$556$	0	0
$557$	−14.8885	−0.630848	−0.315424	−	0.948951i	$-0.602147\pi$
$557$	−14.8885	−0.630848	−0.315424	+	0.948951i	$0.602147\pi$
$558$	0	0
$559$	−10.4721	−0.442924
$560$	0	0
$561$	−10.0000	−0.422200
$562$	0	0
$563$	10.8328	0.456549	0.228274	−	0.973597i	$-0.426692\pi$
$563$	10.8328	0.456549	0.228274	+	0.973597i	$0.426692\pi$
$564$	0	0
$565$	2.76393	0.116279
$566$	0	0
$567$	−24.5967	−1.03297
$568$	0	0
$569$	34.0689	1.42824	0.714121	−	0.700022i	$-0.246827\pi$
$569$	34.0689	1.42824	0.714121	+	0.700022i	$0.246827\pi$
$570$	0	0
$571$	−8.29180	−0.347001	−0.173500	−	0.984834i	$-0.555508\pi$
$571$	−8.29180	−0.347001	−0.173500	+	0.984834i	$0.555508\pi$
$572$	0	0
$573$	−11.0557	−0.461860
$574$	0	0
$575$	17.8885	0.746004
$576$	0	0
$577$	−11.0000	−0.457936	−0.228968	−	0.973434i	$-0.573535\pi$
$577$	−11.0000	−0.457936	−0.228968	+	0.973434i	$0.573535\pi$
$578$	0	0
$579$	−44.5967	−1.85338
$580$	0	0
$581$	−7.23607	−0.300203
$582$	0	0
$583$	−2.36068	−0.0977694
$584$	0	0
$585$	−2.47214	−0.102210
$586$	0	0
$587$	6.47214	0.267134	0.133567	−	0.991040i	$-0.457357\pi$
$587$	6.47214	0.267134	0.133567	+	0.991040i	$0.457357\pi$
$588$	0	0
$589$	40.2492	1.65844
$590$	0	0
$591$	−44.4721	−1.82934
$592$	0	0
$593$	38.3050	1.57300	0.786498	−	0.617593i	$-0.211892\pi$
$593$	38.3050	1.57300	0.786498	+	0.617593i	$0.211892\pi$
$594$	0	0
$595$	2.23607	0.0916698
$596$	0	0
$597$	39.7214	1.62569
$598$	0	0
$599$	−6.23607	−0.254799	−0.127399	−	0.991851i	$-0.540663\pi$
$599$	−6.23607	−0.254799	−0.127399	+	0.991851i	$0.540663\pi$
$600$	0	0
$601$	−22.4721	−0.916658	−0.458329	−	0.888783i	$-0.651552\pi$
$601$	−22.4721	−0.916658	−0.458329	+	0.888783i	$0.651552\pi$
$602$	0	0
$603$	5.88854	0.239800
$604$	0	0
$605$	−9.00000	−0.365902
$606$	0	0
$607$	−8.23607	−0.334292	−0.167146	−	0.985932i	$-0.553455\pi$
$607$	−8.23607	−0.334292	−0.167146	+	0.985932i	$0.553455\pi$
$608$	0	0
$609$	−5.00000	−0.202610
$610$	0	0
$611$	−2.47214	−0.100012
$612$	0	0
$613$	9.52786	0.384827	0.192413	−	0.981314i	$-0.438369\pi$
$613$	9.52786	0.384827	0.192413	+	0.981314i	$0.438369\pi$
$614$	0	0
$615$	3.29180	0.132738
$616$	0	0
$617$	0.639320	0.0257381	0.0128690	−	0.999917i	$-0.495904\pi$
$617$	0.639320	0.0257381	0.0128690	+	0.999917i	$0.495904\pi$
$618$	0	0
$619$	23.1803	0.931697	0.465848	−	0.884865i	$-0.345749\pi$
$619$	23.1803	0.931697	0.465848	+	0.884865i	$0.345749\pi$
$620$	0	0
$621$	−10.0000	−0.401286
$622$	0	0
$623$	−5.52786	−0.221469
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	67.0820	2.67900
$628$	0	0
$629$	−1.70820	−0.0681106
$630$	0	0
$631$	−27.4164	−1.09143	−0.545715	−	0.837971i	$-0.683742\pi$
$631$	−27.4164	−1.09143	−0.545715	+	0.837971i	$0.683742\pi$
$632$	0	0
$633$	59.1935	2.35273
$634$	0	0
$635$	−5.76393	−0.228735
$636$	0	0
$637$	−2.47214	−0.0979496
$638$	0	0
$639$	−12.9443	−0.512067
$640$	0	0
$641$	14.0000	0.552967	0.276483	−	0.961019i	$-0.410831\pi$
$641$	14.0000	0.552967	0.276483	+	0.961019i	$0.410831\pi$
$642$	0	0
$643$	−26.2361	−1.03465	−0.517325	−	0.855789i	$-0.673072\pi$
$643$	−26.2361	−1.03465	−0.517325	+	0.855789i	$0.673072\pi$
$644$	0	0
$645$	−18.9443	−0.745930
$646$	0	0
$647$	−31.6525	−1.24439	−0.622194	−	0.782863i	$-0.713758\pi$
$647$	−31.6525	−1.24439	−0.622194	+	0.782863i	$0.713758\pi$
$648$	0	0
$649$	−4.47214	−0.175547
$650$	0	0
$651$	−30.0000	−1.17579
$652$	0	0
$653$	−29.3607	−1.14897	−0.574486	−	0.818514i	$-0.694798\pi$
$653$	−29.3607	−1.14897	−0.574486	+	0.818514i	$0.694798\pi$
$654$	0	0
$655$	−8.94427	−0.349482
$656$	0	0
$657$	26.8328	1.04685
$658$	0	0
$659$	−16.0689	−0.625955	−0.312977	−	0.949761i	$-0.601326\pi$
$659$	−16.0689	−0.625955	−0.312977	+	0.949761i	$0.601326\pi$
$660$	0	0
$661$	20.8885	0.812470	0.406235	−	0.913769i	$-0.366841\pi$
$661$	20.8885	0.812470	0.406235	+	0.913769i	$0.366841\pi$
$662$	0	0
$663$	2.76393	0.107342
$664$	0	0
$665$	−15.0000	−0.581675
$666$	0	0
$667$	−4.47214	−0.173162
$668$	0	0
$669$	5.52786	0.213720
$670$	0	0
$671$	−61.3050	−2.36665
$672$	0	0
$673$	20.5836	0.793439	0.396720	−	0.917940i	$-0.370149\pi$
$673$	20.5836	0.793439	0.396720	+	0.917940i	$0.370149\pi$
$674$	0	0
$675$	−8.94427	−0.344265
$676$	0	0
$677$	26.0000	0.999261	0.499631	−	0.866239i	$-0.333469\pi$
$677$	26.0000	0.999261	0.499631	+	0.866239i	$0.333469\pi$
$678$	0	0
$679$	12.7639	0.489835
$680$	0	0
$681$	46.1803	1.76963
$682$	0	0
$683$	−17.3475	−0.663785	−0.331892	−	0.943317i	$-0.607687\pi$
$683$	−17.3475	−0.663785	−0.331892	+	0.943317i	$0.607687\pi$
$684$	0	0
$685$	17.0000	0.649537
$686$	0	0
$687$	−8.29180	−0.316352
$688$	0	0
$689$	0.652476	0.0248573
$690$	0	0
$691$	17.4164	0.662551	0.331276	−	0.943534i	$-0.392521\pi$
$691$	17.4164	0.662551	0.331276	+	0.943534i	$0.392521\pi$
$692$	0	0
$693$	−20.0000	−0.759737
$694$	0	0
$695$	0.944272	0.0358183
$696$	0	0
$697$	−1.47214	−0.0557611
$698$	0	0
$699$	7.23607	0.273693
$700$	0	0
$701$	20.4721	0.773222	0.386611	−	0.922243i	$-0.373646\pi$
$701$	20.4721	0.773222	0.386611	+	0.922243i	$0.373646\pi$
$702$	0	0
$703$	11.4590	0.432184
$704$	0	0
$705$	−4.47214	−0.168430
$706$	0	0
$707$	26.1803	0.984613
$708$	0	0
$709$	30.8885	1.16004	0.580022	−	0.814601i	$-0.303044\pi$
$709$	30.8885	1.16004	0.580022	+	0.814601i	$0.303044\pi$
$710$	0	0
$711$	−17.4164	−0.653166
$712$	0	0
$713$	−26.8328	−1.00490
$714$	0	0
$715$	5.52786	0.206730
$716$	0	0
$717$	49.4721	1.84757
$718$	0	0
$719$	−19.8197	−0.739149	−0.369574	−	0.929201i	$-0.620496\pi$
$719$	−19.8197	−0.739149	−0.369574	+	0.929201i	$0.620496\pi$
$720$	0	0
$721$	−26.1803	−0.975007
$722$	0	0
$723$	−13.2918	−0.494327
$724$	0	0
$725$	−4.00000	−0.148556
$726$	0	0
$727$	27.7771	1.03020	0.515098	−	0.857132i	$-0.327756\pi$
$727$	27.7771	1.03020	0.515098	+	0.857132i	$0.327756\pi$
$728$	0	0
$729$	−7.00000	−0.259259
$730$	0	0
$731$	8.47214	0.313353
$732$	0	0
$733$	−11.8885	−0.439113	−0.219557	−	0.975600i	$-0.570461\pi$
$733$	−11.8885	−0.439113	−0.219557	+	0.975600i	$0.570461\pi$
$734$	0	0
$735$	−4.47214	−0.164957
$736$	0	0
$737$	−13.1672	−0.485019
$738$	0	0
$739$	48.0689	1.76824	0.884121	−	0.467258i	$-0.154758\pi$
$739$	48.0689	1.76824	0.884121	+	0.467258i	$0.154758\pi$
$740$	0	0
$741$	−18.5410	−0.681121
$742$	0	0
$743$	−1.88854	−0.0692840	−0.0346420	−	0.999400i	$-0.511029\pi$
$743$	−1.88854	−0.0692840	−0.0346420	+	0.999400i	$0.511029\pi$
$744$	0	0
$745$	14.6525	0.536825
$746$	0	0
$747$	−6.47214	−0.236803
$748$	0	0
$749$	12.8885	0.470937
$750$	0	0
$751$	−21.4164	−0.781496	−0.390748	−	0.920498i	$-0.627784\pi$
$751$	−21.4164	−0.781496	−0.390748	+	0.920498i	$0.627784\pi$
$752$	0	0
$753$	17.1115	0.623576
$754$	0	0
$755$	9.23607	0.336135
$756$	0	0
$757$	37.0000	1.34479	0.672394	−	0.740193i	$-0.265266\pi$
$757$	37.0000	1.34479	0.672394	+	0.740193i	$0.265266\pi$
$758$	0	0
$759$	−44.7214	−1.62328
$760$	0	0
$761$	−2.41641	−0.0875947	−0.0437974	−	0.999040i	$-0.513946\pi$
$761$	−2.41641	−0.0875947	−0.0437974	+	0.999040i	$0.513946\pi$
$762$	0	0
$763$	1.05573	0.0382199
$764$	0	0
$765$	2.00000	0.0723102
$766$	0	0
$767$	1.23607	0.0446318
$768$	0	0
$769$	16.7639	0.604523	0.302261	−	0.953225i	$-0.402258\pi$
$769$	16.7639	0.604523	0.302261	+	0.953225i	$0.402258\pi$
$770$	0	0
$771$	−2.23607	−0.0805300
$772$	0	0
$773$	45.0132	1.61901	0.809505	−	0.587113i	$-0.199735\pi$
$773$	45.0132	1.61901	0.809505	+	0.587113i	$0.199735\pi$
$774$	0	0
$775$	−24.0000	−0.862105
$776$	0	0
$777$	−8.54102	−0.306407
$778$	0	0
$779$	9.87539	0.353823
$780$	0	0
$781$	28.9443	1.03571
$782$	0	0
$783$	2.23607	0.0799106
$784$	0	0
$785$	22.1803	0.791650
$786$	0	0
$787$	1.52786	0.0544625	0.0272312	−	0.999629i	$-0.491331\pi$
$787$	1.52786	0.0544625	0.0272312	+	0.999629i	$0.491331\pi$
$788$	0	0
$789$	15.0000	0.534014
$790$	0	0
$791$	−6.18034	−0.219748
$792$	0	0
$793$	16.9443	0.601709
$794$	0	0
$795$	1.18034	0.0418623
$796$	0	0
$797$	−49.1246	−1.74008	−0.870042	−	0.492978i	$-0.835908\pi$
$797$	−49.1246	−1.74008	−0.870042	+	0.492978i	$0.835908\pi$
$798$	0	0
$799$	2.00000	0.0707549
$800$	0	0
$801$	−4.94427	−0.174697
$802$	0	0
$803$	−60.0000	−2.11735
$804$	0	0
$805$	10.0000	0.352454
$806$	0	0
$807$	36.1803	1.27361
$808$	0	0
$809$	−7.23607	−0.254407	−0.127203	−	0.991877i	$-0.540600\pi$
$809$	−7.23607	−0.254407	−0.127203	+	0.991877i	$0.540600\pi$
$810$	0	0
$811$	44.5410	1.56405	0.782023	−	0.623249i	$-0.214188\pi$
$811$	44.5410	1.56405	0.782023	+	0.623249i	$0.214188\pi$
$812$	0	0
$813$	16.0557	0.563099
$814$	0	0
$815$	3.05573	0.107037
$816$	0	0
$817$	−56.8328	−1.98833
$818$	0	0
$819$	5.52786	0.193159
$820$	0	0
$821$	3.34752	0.116829	0.0584147	−	0.998292i	$-0.481395\pi$
$821$	3.34752	0.116829	0.0584147	+	0.998292i	$0.481395\pi$
$822$	0	0
$823$	−12.1803	−0.424580	−0.212290	−	0.977207i	$-0.568092\pi$
$823$	−12.1803	−0.424580	−0.212290	+	0.977207i	$0.568092\pi$
$824$	0	0
$825$	−40.0000	−1.39262
$826$	0	0
$827$	−8.58359	−0.298481	−0.149240	−	0.988801i	$-0.547683\pi$
$827$	−8.58359	−0.298481	−0.149240	+	0.988801i	$0.547683\pi$
$828$	0	0
$829$	−3.00000	−0.104194	−0.0520972	−	0.998642i	$-0.516591\pi$
$829$	−3.00000	−0.104194	−0.0520972	+	0.998642i	$0.516591\pi$
$830$	0	0
$831$	2.23607	0.0775683
$832$	0	0
$833$	2.00000	0.0692959
$834$	0	0
$835$	22.7082	0.785850
$836$	0	0
$837$	13.4164	0.463739
$838$	0	0
$839$	−12.2918	−0.424360	−0.212180	−	0.977231i	$-0.568056\pi$
$839$	−12.2918	−0.424360	−0.212180	+	0.977231i	$0.568056\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	0	0
$843$	56.7082	1.95313
$844$	0	0
$845$	11.4721	0.394653
$846$	0	0
$847$	20.1246	0.691490
$848$	0	0
$849$	37.2361	1.27794
$850$	0	0
$851$	−7.63932	−0.261873
$852$	0	0
$853$	−49.4721	−1.69389	−0.846947	−	0.531678i	$-0.821562\pi$
$853$	−49.4721	−1.69389	−0.846947	+	0.531678i	$0.821562\pi$
$854$	0	0
$855$	−13.4164	−0.458831
$856$	0	0
$857$	−52.0000	−1.77629	−0.888143	−	0.459567i	$-0.848005\pi$
$857$	−52.0000	−1.77629	−0.888143	+	0.459567i	$0.848005\pi$
$858$	0	0
$859$	−0.472136	−0.0161091	−0.00805454	−	0.999968i	$-0.502564\pi$
$859$	−0.472136	−0.0161091	−0.00805454	+	0.999968i	$0.502564\pi$
$860$	0	0
$861$	−7.36068	−0.250851
$862$	0	0
$863$	16.4721	0.560718	0.280359	−	0.959895i	$-0.409546\pi$
$863$	16.4721	0.560718	0.280359	+	0.959895i	$0.409546\pi$
$864$	0	0
$865$	4.18034	0.142136
$866$	0	0
$867$	−2.23607	−0.0759408
$868$	0	0
$869$	38.9443	1.32109
$870$	0	0
$871$	3.63932	0.123314
$872$	0	0
$873$	11.4164	0.386387
$874$	0	0
$875$	20.1246	0.680336
$876$	0	0
$877$	−27.0000	−0.911725	−0.455863	−	0.890050i	$-0.650669\pi$
$877$	−27.0000	−0.911725	−0.455863	+	0.890050i	$0.650669\pi$
$878$	0	0
$879$	32.2361	1.08730
$880$	0	0
$881$	21.3475	0.719216	0.359608	−	0.933103i	$-0.382910\pi$
$881$	21.3475	0.719216	0.359608	+	0.933103i	$0.382910\pi$
$882$	0	0
$883$	−37.1803	−1.25122	−0.625609	−	0.780137i	$-0.715149\pi$
$883$	−37.1803	−1.25122	−0.625609	+	0.780137i	$0.715149\pi$
$884$	0	0
$885$	2.23607	0.0751646
$886$	0	0
$887$	−53.7771	−1.80566	−0.902829	−	0.430000i	$-0.858514\pi$
$887$	−53.7771	−1.80566	−0.902829	+	0.430000i	$0.858514\pi$
$888$	0	0
$889$	12.8885	0.432268
$890$	0	0
$891$	49.1935	1.64804
$892$	0	0
$893$	−13.4164	−0.448963
$894$	0	0
$895$	18.1803	0.607702
$896$	0	0
$897$	12.3607	0.412711
$898$	0	0
$899$	6.00000	0.200111
$900$	0	0
$901$	−0.527864	−0.0175857
$902$	0	0
$903$	42.3607	1.40968
$904$	0	0
$905$	−14.8885	−0.494912
$906$	0	0
$907$	−8.70820	−0.289151	−0.144576	−	0.989494i	$-0.546182\pi$
$907$	−8.70820	−0.289151	−0.144576	+	0.989494i	$0.546182\pi$
$908$	0	0
$909$	23.4164	0.776673
$910$	0	0
$911$	−42.1246	−1.39565	−0.697825	−	0.716268i	$-0.745849\pi$
$911$	−42.1246	−1.39565	−0.697825	+	0.716268i	$0.745849\pi$
$912$	0	0
$913$	14.4721	0.478958
$914$	0	0
$915$	30.6525	1.01334
$916$	0	0
$917$	20.0000	0.660458
$918$	0	0
$919$	−18.0689	−0.596037	−0.298019	−	0.954560i	$-0.596326\pi$
$919$	−18.0689	−0.596037	−0.298019	+	0.954560i	$0.596326\pi$
$920$	0	0
$921$	43.9443	1.44801
$922$	0	0
$923$	−8.00000	−0.263323
$924$	0	0
$925$	−6.83282	−0.224662
$926$	0	0
$927$	−23.4164	−0.769096
$928$	0	0
$929$	46.9443	1.54019	0.770096	−	0.637928i	$-0.220208\pi$
$929$	46.9443	1.54019	0.770096	+	0.637928i	$0.220208\pi$
$930$	0	0
$931$	−13.4164	−0.439705
$932$	0	0
$933$	−11.8328	−0.387389
$934$	0	0
$935$	−4.47214	−0.146254
$936$	0	0
$937$	−19.0557	−0.622524	−0.311262	−	0.950324i	$-0.600752\pi$
$937$	−19.0557	−0.622524	−0.311262	+	0.950324i	$0.600752\pi$
$938$	0	0
$939$	8.54102	0.278726
$940$	0	0
$941$	42.1803	1.37504	0.687520	−	0.726166i	$-0.258699\pi$
$941$	42.1803	1.37504	0.687520	+	0.726166i	$0.258699\pi$
$942$	0	0
$943$	−6.58359	−0.214391
$944$	0	0
$945$	−5.00000	−0.162650
$946$	0	0
$947$	−23.1803	−0.753260	−0.376630	−	0.926364i	$-0.622917\pi$
$947$	−23.1803	−0.753260	−0.376630	+	0.926364i	$0.622917\pi$
$948$	0	0
$949$	16.5836	0.538326
$950$	0	0
$951$	7.88854	0.255804
$952$	0	0
$953$	−56.2492	−1.82209	−0.911046	−	0.412306i	$-0.864724\pi$
$953$	−56.2492	−1.82209	−0.911046	+	0.412306i	$0.864724\pi$
$954$	0	0
$955$	−4.94427	−0.159993
$956$	0	0
$957$	10.0000	0.323254
$958$	0	0
$959$	−38.0132	−1.22751
$960$	0	0
$961$	5.00000	0.161290
$962$	0	0
$963$	11.5279	0.371480
$964$	0	0
$965$	−19.9443	−0.642029
$966$	0	0
$967$	36.7639	1.18225	0.591124	−	0.806581i	$-0.298684\pi$
$967$	36.7639	1.18225	0.591124	+	0.806581i	$0.298684\pi$
$968$	0	0
$969$	15.0000	0.481869
$970$	0	0
$971$	−11.8754	−0.381099	−0.190550	−	0.981678i	$-0.561027\pi$
$971$	−11.8754	−0.381099	−0.190550	+	0.981678i	$0.561027\pi$
$972$	0	0
$973$	−2.11146	−0.0676902
$974$	0	0
$975$	11.0557	0.354067
$976$	0	0
$977$	−8.58359	−0.274613	−0.137307	−	0.990529i	$-0.543845\pi$
$977$	−8.58359	−0.274613	−0.137307	+	0.990529i	$0.543845\pi$
$978$	0	0
$979$	11.0557	0.353343
$980$	0	0
$981$	0.944272	0.0301483
$982$	0	0
$983$	41.7771	1.33248	0.666241	−	0.745736i	$-0.267902\pi$
$983$	41.7771	1.33248	0.666241	+	0.745736i	$0.267902\pi$
$984$	0	0
$985$	−19.8885	−0.633702
$986$	0	0
$987$	10.0000	0.318304
$988$	0	0
$989$	37.8885	1.20479
$990$	0	0
$991$	−10.4033	−0.330470	−0.165235	−	0.986254i	$-0.552838\pi$
$991$	−10.4033	−0.330470	−0.165235	+	0.986254i	$0.552838\pi$
$992$	0	0
$993$	−61.8328	−1.96221
$994$	0	0
$995$	17.7639	0.563155
$996$	0	0
$997$	21.9443	0.694982	0.347491	−	0.937683i	$-0.387034\pi$
$997$	21.9443	0.694982	0.347491	+	0.937683i	$0.387034\pi$
$998$	0	0
$999$	3.81966	0.120849

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.a.e.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.a.e.1.1	✓	2	1.1	even	1	trivial

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Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4012.a (trivial)

\(f(q)\)	\(=\)	\(q-2.23607 q^{3} -1.00000 q^{5} +2.23607 q^{7} +2.00000 q^{9} +O(q^{10})\)	\(q-2.23607 q^{3} -1.00000 q^{5} +2.23607 q^{7} +2.00000 q^{9} -4.47214 q^{11} +1.23607 q^{13} +2.23607 q^{15} -1.00000 q^{17} +6.70820 q^{19} -5.00000 q^{21} -4.47214 q^{23} -4.00000 q^{25} +2.23607 q^{27} +1.00000 q^{29} +6.00000 q^{31} +10.0000 q^{33} -2.23607 q^{35} +1.70820 q^{37} -2.76393 q^{39} +1.47214 q^{41} -8.47214 q^{43} -2.00000 q^{45} -2.00000 q^{47} -2.00000 q^{49} +2.23607 q^{51} +0.527864 q^{53} +4.47214 q^{55} -15.0000 q^{57} +1.00000 q^{59} +13.7082 q^{61} +4.47214 q^{63} -1.23607 q^{65} +2.94427 q^{67} +10.0000 q^{69} -6.47214 q^{71} +13.4164 q^{73} +8.94427 q^{75} -10.0000 q^{77} -8.70820 q^{79} -11.0000 q^{81} -3.23607 q^{83} +1.00000 q^{85} -2.23607 q^{87} -2.47214 q^{89} +2.76393 q^{91} -13.4164 q^{93} -6.70820 q^{95} +5.70820 q^{97} -8.94427 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} + 4 q^{9}+O(q^{10}) \) 2 * q - 2 * q^5 + 4 * q^9	\( 2 q - 2 q^{5} + 4 q^{9} - 2 q^{13} - 2 q^{17} - 10 q^{21} - 8 q^{25} + 2 q^{29} + 12 q^{31} + 20 q^{33} - 10 q^{37} - 10 q^{39} - 6 q^{41} - 8 q^{43} - 4 q^{45} - 4 q^{47} - 4 q^{49} + 10 q^{53} - 30 q^{57} + 2 q^{59} + 14 q^{61} + 2 q^{65} - 12 q^{67} + 20 q^{69} - 4 q^{71} - 20 q^{77} - 4 q^{79} - 22 q^{81} - 2 q^{83} + 2 q^{85} + 4 q^{89} + 10 q^{91} - 2 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 + 4 * q^9 - 2 * q^13 - 2 * q^17 - 10 * q^21 - 8 * q^25 + 2 * q^29 + 12 * q^31 + 20 * q^33 - 10 * q^37 - 10 * q^39 - 6 * q^41 - 8 * q^43 - 4 * q^45 - 4 * q^47 - 4 * q^49 + 10 * q^53 - 30 * q^57 + 2 * q^59 + 14 * q^61 + 2 * q^65 - 12 * q^67 + 20 * q^69 - 4 * q^71 - 20 * q^77 - 4 * q^79 - 22 * q^81 - 2 * q^83 + 2 * q^85 + 4 * q^89 + 10 * q^91 - 2 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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